Core Graduate Courses | Other Courses
Mathematics graduate students can take suitable graduate courses in other departments (CS, Physics, Chemistry, Eng. etc. ) to satisfy their course credit requirement. Two-thirds of the course requirements for each degree should be in the Mathematics Department.
Please contact the Graduate Office early on if you are looking for enrollment in popular courses, as they fill up fast.
CORE COURSES
MAT1000HF (MAT457H1F)
REAL ANALYSIS I
Measure Theory: Lebesque measure and integration, convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem.
Functional Analysis: Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^p-spaces, Holder and Minkowski inequalities.
Textbook: Gerald Folland, Real Analysis: Modern Techniques and their Applications, Wiley
References:
Elias Stein and Rami Shakarchi: Measure Theory, Integration, and Hilbert Spaces
Elliott H. Lieb and Michael Loss: Analysis AMS Graduate Texts in Mathematics, 14 (either edition)
H.L. Royden: Real Analysis, Macmillan, 1988.
A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis, 1975.
MAT1001HS (MAT458H1S)
REAL ANALYSIS II
Basic Functional Analysis: Banach spaces, Hilbert space, Hahn Banach theorem, open mapping theorem, closed graph theorem, uniform boundedness principle, Alaoglu's theorem, Frechet spaces.
Fourier Analysis: Fourier series and transforms, Fourier inversion and Plancherel formula, estimates and convergence results, more topological vector spaces, Schwartz space, distributions.
Spectral theory: spectral theorem for bounded self-adjoint operators, specializations to compact operators and/or extensions to unbounded operators, as time permits.
Textbooks:
G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley; W. Rudin, Functional Analysis, Second Edition, Indian Edition (if available; the book is hard to get, although there is a pdf on line).
MAT1002HS (MAT454H1S)
COMPLEX ANALYSIS
Review of holomorphic and harmonic functions. Topology of a space of holomorphic functions: series and infinite products, Weierstrass -function, Weierstrass and Mittag-Leffler theorems. Normal families (compact subsets of H()). Conformal mappings: Riemann mapping theorem, Schwarz-Christoffel formula. Riemann surfaces: Riemann surface associated with an elliptic curve, inversion of an elliptic integral, Abel’s theorem. Analytic continuation, Sheaf of holomorphic functions, Monodromy theorem, Little Picard theorem.
Recommended prerequisites:
A first course in complex analysis and a course in real analysis. Measure theory is not required.
Textbook:
L. Ahlfors: Complex Analysis, 3rd Edition, AMS Chelsea
Recommended References:
H. Cartan, Elementary Theory of Analytic Functions of One or Several Complex Variables, Dover;T. Gamelin, Complex Analysis, Springer; O. Ivrii, The Bierstone Lectures on Complex Analysis, course notes online.
MAT1060HF
PARTIAL DIFFERENTIAL EQUATIONS I
I. Uriarte-Tuero
(View Timetable)
This is a basic introduction to partial differential equations as they arise in physics, geometry and optimization. It is meant to be accessible to beginners with little or no prior knowledge of the field. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' basic bag of tools. A key theme will be the development of techniques for studying non-smooth solutions to these equations.
Textbook:
Lawrence C. Evans, Partial Differential Equations, 2nd Edition, AMS GSM19, ISBN 978-1-4704-6942-9.
MAT1061HS
PARTIAL DIFFERENTIAL EQUATIONS II
This course will consider a range of mostly nonlinear partial differential equations, including elliptic and parabolic PDE, as well as hyperbolic and other nonlinear wave equations. In order to study these equations, we will develop a variety of methods, including variational techniques, and fixed point theorems. One important theme will be the relationship between variational questions, such as critical Sobolev exponents, and issues related to nonlinear evolution equations, such as finite-time blowup of solutions and/or long-time asymptotics.
Prerequisites:
Familiarity with Sobolev and other function spaces, and in particular with fundamental embedding and compactness theorems. Other topics in PDE will also be discussed.
Reference:
Robert C. McOwen, Partial Differential Equations, 2nd edition, ISBN 0-13-009335-1.
MAT1100HF
ALGEBRA I
Basic notions of linear algebra: brief recollection. The language of Hom spaces and the corresponding canonical isomorphisms. Tensor product of vector spaces.
Group Theory: Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations.
Ring Theory:
Rings, ideals, Euclidean domains, principal ideal domains, and unique factorization domains.
Modules:
Modules and algebras over a ring, tensor products, modules over a principal ideal domain
Recommended prerequisites are a full year undergraduate course in Linear Algebra and one term of an introductory undergraduate course in higher algebra, covering, at least, basic group theory. While this material will be reviewed in the course, it will be done at "high speed", assuming that you have already some familiarity with the basics. You will be very well prepared indeed, if you have no difficulties reading and understanding the book, listed here under "Other References", M. Artin: Algebra that the author wrote for his undergraduate algebra courses at MIT.
Textbooks:
Lang: Algebra, 3rd edition
Dummit and Foote: Abstract Algebra, 2nd Edition
Other References:
Jacobson: Basic Algebra, Volumes I and II
Cohn: Basic Algebra
M. Artin: Algebra.
MAT1101HS
ALGEBRA II
Fields: Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois Theory, solution of equations by radicals.
Commutative Rings:
Noetherian rings, Hilbert basis theorem, invariant theory, Hilbert Nullstellensatz, primary decomposition, affine algebraic varieties. Structure of semisimple algebras, application to representation theory of finite groups.
Textbooks:
Dummit and Foote: Abstract Algebra, 3rd Edition
Lang: Algebra, 3rd Edition
Other References:
Jacobson: Basic Algebra, Volumes I and II
Cohn: Basic Algebra
M. Artin: Algebra.
MAT1300HF
DIFFERENTIAL TOPOLOGY
Local differential geometry:
The differential, the inverse function theorem, smooth manifolds, the tangent space, immersions and submersions, regular points, transversality, Sard's theorem, the Whitney embedding theorem, smooth approximation, tubular neighborhoods, the Brouwer fixed point theorem.
Differential forms:
Exterior algebra, forms, pullbacks, integration, Stokes' theorem, div grad curl and all, Lagrange's equation and Maxwell's equations, homotopies and Poincare's lemma, linking numbers.
Prerequisites:
linear algebra; vector calculus; point set topology
Textbook:
John M. Lee: Introduction to Smooth Manifolds
MAT1301HS
ALGEBRAIC TOPOLOGY
Fundamental groups:
Paths and homotopies, the fundamental group, coverings and the fundamental group of the circle, Van-Kampen's theorem, the general theory of covering spaces.
Homology:
Simplices and boundaries, prisms and homotopies, abstract nonsense and diagram chasing, axiomatics, degrees, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, salad bowls and Borsuk-Ulam, cohomology and de-Rham's theorem, products.
Textbook:
Allen Hatcher, Algebraic Topology
Recommended Textbooks:
Munkres, Topology
Munkres, Algebraic Topology
MAT1600HF
MATHEMATICAL PROBABILITY I
The class will cover classical limit theorems for sums of independent random variables, such as the Law of Large Numbers and Central Limit Theorem, conditional distributions and martingales, metrics on probability measures.
Textbook:
Lecture notes and a list of recommended books will be provided.
Recommended prerequisite:
Real Analysis I.
MAT1601HS
MATHEMATICAL PROBABILITY II
The class will cover some of the following topics: Brownian motion and examples of functional central limit theorems, Gaussian processes, Poisson processes, Markov chains, exchangeability.
Recommended prerequisites:
Real Analysis I and Probability I.
MAT1850HF
LINEAR ALGEBRA AND OPTIMIZATION
A. Stinchcombe
(View Timetable)
This course will develop advanced methods in linear algebra and introduce the theory of optimization. On the linear algebra side, we will study important matrix factorizations (e.g. LU, QR, SVD), matrix approximations (both deterministic and randomized), convergence of iterative methods, and spectral theorems. On the optimization side, we will introduce the finite element method, linear programming, gradient methods, and basic convex optimization. The course will be focused on fundamental theory, but appropriate illustrative applications may be chosen by the instructor.
Other Courses
MAT1005HF
Fourier Analysis
Introduction to basic objects and tools in Fourier/Harmonic analysis, such as the Maximal function, the Hilbert transform, Singular integrals, Calderón-Zygmund theory etc.
Basics of Littlewood-Paley theory and elements of Paradifferential calculus.
Applications to nonlinear partial differential equations in fluid dynamics (Euler/Navier-Stokes) and/or quantum mechanics (nonlinear Schrodinger)
Some possible additional topics (time permitting, or upon request): Oscillatory integrals, the theory of Pseudo/Para-differential operators, Coifman-Meyer Theory, applications of Fourier analytical tools to the study of the long-time behavior of semilinear and quasilinear evolution PDEs.
Fourier series. Maximal funcion. Hilbert transform and Calder\'on-Zygmund operators. H^1 and BMO. Weighted inequalities. Time permitting, Littlewood-Paley theory and/or T1 theorems.
Textbook:
uoandikoetxea "Fourier Analysis"
Prerequisites:
AT457. Desirable MAT458.
MAT1126HF
Lie Groups and Fluid Dynamics: Topological Fluid Dynamics
This course deals with various problems in geometry, Lie theory, and Hamiltonian systems, motivated by hydrodynamics and magnetohydrodynamics. We discuss the dynamics of an ideal fluid from the group-theoretic and Hamiltonian points of view. We cover geometry of conservation laws of the Euler equation, point vortex approximations, topology of steady flows and their nonlinear stability, relation of the energy and helicity of vector fields, geometry of diffeomorphism groups, relation to vortex sheets, as well descriptions of magnetohydrodynamics and of the Korteweg-de Vries equation in the Lie group framework.
Syllabus:
- Main notions in Lie groups and Lie algebras.
- The Euler equation of an ideal fluid as a geodesic flow. General Euler-Arnold equations.
- The Hamiltonian framework for the Euler equations: Equations on the dual Lie algebra, Poisson structures.
- Examples: Ideal hydrodynamics, Magnetohydrodynamics (MHD), the Korteweg-deVries and Camassa-Holm equations (on Virasoro groups), β-plane equation in meteorology, Landau-Lifschits equation.
- Conservation laws for fluids: First integrals for ideal and compressible fluids. Point vortex approximations in 2D. Vortex filament equation and the Marsden-Weinstein structures on the spaces of curves.
- Steady solutions in 2D and 3D. Bernoulli function, related variational problems, restrictions on topology of steady solutions, Arnold’s stability criterion.
- Topology bounds the energy of a field. An ergodic interpretation of helicity (asymptotic Hopf invariant), energy estimates, Sakharov–Zeldovich problem.
- Other applications (time permitted): Differential geometry of diffeomorphism groups: Otto calculus and optimal transport, geometry of vortex sheets and shock waves.
V.Arnold and B.Khesin: Topological Methods in Hydrodynamics, Appl. Math. Series, v. 125, Springer-Verlag, 1998, Second extended edition: 2021.
J.Marsden and T.Ratiu: Introduction to Mechanics and Symmetry, Texts in Applied Math., v. 17, Springer-Verlag, 1994/1999.
MAT1128HF
Topics in Probability: Modern discrete probability??
Brief course outline: The goal of the course is to give an introduction to fundamental models and techniques in graduate-level modern discrete probability. Topics include random walks, percolation, spin systems, random graphs and networks, and first and last passage percolation. The emphasis in on covering common or important techniques, rather than covering any one area in particular depth.
Very roughly, the course will look something like MAT 589, with more of an emphasis on lattice models (percolation, Ising model, first and last passage percolation).
MAT1128HS
Topics in Mathematical Physics: Statistical mechanics of lattice models
A rigorous introduction to the statistical mechanics of spin systems on the lattice, such as the Ising and O(N) models. Phase transitions, high temperature and low temperature expansions. Behaviour at the critical point. Correlation inequalities. Triviality in high dimensions.
Prerequisites:
Graduate probability.
Textbooks:
S. Friedli. and Y. Velenik, Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Roland Bauerschmidt's lecture notes on spin systems.
MAT1190HF
Algebraic Geometry I
This course will be an introduction to the theory of schemes, following chapter 2 in Hartshorne. The following topics will be covered:
- Sheaves and locally ringed spaces
- Schemes
- Quasi-coherent sheaves
- Open and closed immersions
- Fibre products
- Separated and proper morphisms
- Valuative criterion
- Projective morphisms
- Kähler differentials
Prerequisites:
Algebra 1 & 2, in particular we expect familiarity with rings, ideals, modules, tensor products.
Textbook:
Hartshorne, Algebraic geometry
Complementary reading of Vakil's book is highly encouraged.
MAT1190HS
Algebraic Geometry II
This is the sequel to Algebraic Geometry I, which will cover Chapter II of Hartshorne.
This course will cover Chapter III of Hartshorne, including derived functors, sheaf cohomology, Cech cohomology, the cohomology of projective space and other examples, Ext groups, Serre duality, higher direct images, flat and smooth morphisms, formal GAGA, the semicontinuity theorem, and cohomology and base change. It is meant as the second-half of a year long course meant to provide graduate students with the necessary background to do research in algebraic geometry.
Prerequisites:
Algebraic Geometry I
Textbook:
Algebraic Geometry, Hartshorne
MAT1210HS
Topics in Number Theory: Motives, Periods and Transcendence
The transcendence (or even irrationality) of special values of L-functions is a central theme in number theory. Euler's theorem that the Riemann zeta function at positive even integers is a rational multiple of a power of a single transcendental number, namely π, is an early success story. But Euler also tried to find a similar theorem about the values of the zeta function at positive odd integers and was unable to do so.
In this course, we will describe the modern perspective on this theme, namely to interpret these values as periods of certain motives (or mixed motives). We will give an exposition of the work of Kontsevich and Zagier on periods, some of the work of Francis Brown on multiple zeta values and motives, and Grothendieck's period conjecture.
Finally, we will give an exposition of recent work on some special values of Dirichlet L-functions, including the (apparently non-motivic) irrationality result by Calegari, Dimitrov and Tang, as well as the (motivic) work of Eskandari, Murty and Nemoto on Catalan's constant.
Prerequisites and Evaluation
We will assume basic number theory and algebraic geometry, but will attempt to make the content as self-contained as possible. The field is just opening up now and the aim of the course will really be to get more students up to speed on these ideas. Evaluation will be based on attendance and class participation.
References
- F. Brown, Irrationality proofs of zeta values, moduli spaces and dinner parties, Moscow Journal of Combinatorics and Number Theory, 6(2016), 102-165.
- F. Calegari, V. Dimitrov and Y. Tang, The linear independence of 1, ς(2) and L(2, χ-3), arXiv:2408.15403 [math.NT], 2024.
- P. Eskandari and V. Kumar Murty, On unipotent radicals of motivic Galois groups, Al-gebra and Number Theory, 17(2023), 165-214.
- P. Eskandari, V. Kumar Murty and Y. Nemoto, Mixed motives and the Catalan constant, arXiv:2510.20648 [math.NT], 2025.
- M. Kontsevich and D. Zagier, Periods, IHES Publication, 2001.
MAT1211HS
Topics in Number Theory II: Class Field Theory
This is an introduction to local and global class field theory. Class field theory is a crowning achievement of classical algebraic number theory and also one of the starting points of the Langlands program.
Tentative plan: local class field theory, proved using Lubin-Tate theory of formal groups. Group and Galois cohomology. Statements of global class field theory, both classical and modern (adelic). Some applications. Proofs to be discussed as much as time permits.
Prerequisites:
Algebraic number theory as in MAT1200 (MAT415).
This course would make the most sense as a sequel to Algebraic Number Theory MAT1200, i.e. in Winter term, assuming MAT1200 is offered in the Fall.
MAT1304HF
Topics in Combinatorics: Extremal Combinatorics
L. Gishboliner
(View Timetable)
This course serves as an introduction to extremal graph theory and Ramsey theory. The second half of the course covers more advanced topics, and serves as a good preparation for doing research in these areas.
Syllabus:
- Ramsey's theorem for graphs, lower bounds via the probabilistic method.
- Hypergraph Ramsey numbers, stepping down and stepping up.
- Ordered Ramsey numbers, the Erdos-Szekeres lemma and the cups-caps theorem.
- Turan's theorem.
- Extremal theory for bipartite graphs: Kovari-Sos-Turan theorem; lower bound constructions; extremal numbers of trees, paths and cycles; dependent random choice.
- The container method.
- The regularity method.
- Off-diagonal Ramsey numbers: The Ajtai-Komlos-Szemeredi theorem, R(4,t).
Additional topics if time permits: quasirandomness, the Erdos-Hajnal conjecture, random algebraic constructions.
Prerequisites:
MAT344. MAT332 is recommended. There are no other hard prerequisites, but mathematical maturity is required.
MAT1304HS
Topics in Combinatorics: Combinatorics/ discrete mathematics
Over the past few decades, algebraic methods have become extremely powerful tools in several areas of combinatorics and computer science. Many recent and important advances in these fields rely on surprisingly simple properties of polynomials.
In this course, we will explore a variety of interesting and often surprising applications of linear algebra and polynomial methods to combinatorics, discrete geometry, complexity theory, cryptography, and algorithm design. All necessary algebraic tools will be developed throughout the course as needed.
The main prerequisite for the course is mathematical maturity. Familiarity with discrete mathematics, algorithms, and linear algebra will be helpful but is not strictly required. Students with an interest in discrete mathematics and/or theoretical computer science are especially encouraged to enroll.
MAT1307HF/ CSC2406H
Mathematical Methods in the Theory of Computation
This course will expose students to mathematical methods that play a role in recent advances in theoretical computer science. The course will use (and develop) tools from combinatorics, probability, algebra and analysis, and will discuss applications to modern topics in the theory of computation.
MAT1309HS
Geometric Inequalities: Geometric Measure Theory
Y. Liokumovich
(View Timetable)
Rectifiable sets and measures, integral currents: mass, boundary, compactness, and existence of area-minimizing currents. Varifolds and first variation of area, stationary varifolds, and monotonicity formulas. Regularity theory for area-minimizing submanifolds and an introduction to min–max minimal submanifolds. Main reference: Leon Simon, Lectures on Geometric Measure Theory.
MAT1312HS
Topics in Geometry: Introduction to Motives/Motiivic Homotopy Theory
This course offers a practical introduction to motivic homotopy theory, focusing on the theory of non-A1-invariant motivic homotopy theory introduced recently by Annala–Hoyois–Iwasa. Emphasis is put on how to work with motives in practice. The aim is also to make it clear from the beginning that motivic methods apply to essentially all cohomology theories that occur in algebraic geometry.
Prerequisites:
We assume basic knowledge of algebraic geometry, algebraic topology (e.g. the basic graduate sequences on these subjects), and category theory (functors, adjunctions, limits, colimits, etc). We will use infinity-categorical methods, but students will not need to master the foundations, but are recommended to take standard results, such as those that can be found in Higher Topos Theory, on faith.
Textbook:
No book but I will write lecture notes.
Course outline:
- Categorical toolbox: (Stable) infinity categories
- Geometric toolbox: Nisnevich topology and smooth blowups
- The category of motivic spectra and its universal property
- Examples of motivic spectra, realization functors
- Symmetric monoidality of the Thom spectrum construction
- P-homotopy invariance and deflatability
- Infinite excision, geometric models for BGL_n
- Oriented motivic spectra
- Gysin maps and motivic Poincaré duality
- Punchline. Options: a. Prismatic Steenrod operations and Solution to Tate’s conjecture on Brauer groups (Carmeli–Feng); b. A1-colocalization, logarithmic cohomology is an invariant of the open part; c. Resolvable motives (Annala–Pstragowski), du Bois theory in positive characteristic (analogue of Park–Popa), Zeta functions (Hyslop)
MAT1318HS
Seminar in Geometry and Topology: Invitation to Alexandrov geometry: CAT(0) spaces
The course will cover basics of CAT(0) spaces, globalization theorem, Reshetnyak’s gluing theorem, Reshetnyak’s majorization theorem, application to billiards, polyhedral CAT(0) spaces construction of exotic aspherical manifolds. If time allws we will also cover 2-covexity. The course will be based on my book with Petrunin and Alexander: Invitation to Alexandrov geometry: CAT(0) spaces.
MAT1341HF
Topics in Differential Geometry: Moduli spaces of flat bundles over surfaces
We shall discuss the representation variety M_G(S) of homomorphisms from the fundamental group of a compact oriented surface S with values in a Lie group G. Equivalently, this is the moduli space of flat G-bundles over the surface S. When the Lie algebra of G comes with an invariant inner product, then the moduli space acquires a symplectic structure, obtained by Atiyah-Bott through an infinite-dimensional symplectic reduction of a space of connections, with curvature as a moment map. The theory of quasi-Hamiltonian spaces gives a finite-dimensional construction of the moduli space, by reduction of a space with a G-valued moment map. As special cases, we shall discuss Teichmueller spaces of hyperbolic structures, as well as moduli spaces of projective structures within this framework. The course will start out with a quick introduction to the topology of surfaces, and some background on G-bundles and gauge theory. We will then proceed to discussion of the symplectic structure. Other topics will be included depending on student's interests.
Prerequsites:
ood working knowledge of differential geometry, including differential forms and the relation between Lie groups and Lie algebras.
References:
F. Labourie: Lectures on representations of surface groups, 2017
E. Meinrenken: Introduction to moduli spaces and Dirac geometry, 2025
Score/ 0 pts
MAT1502HF
Topics in Geometric Analysis: Analysis and Geometry of Metric Measure Spaces
Nonsmooth objects arise ubiquitously in nature, applications, and as limits of smooth objects. To analyze them requires ideas from geometry which apply outside the traditional framework of smooth manifolds. The rapidly developing areas of metric and metric-measure geometry provide an ideal setting for this, enabling powerful ideas to be simplified and generalized.
This course will be an introduction to these topics. It will expose how various notions of curvature (e.g. sectional and Ricci) can be estimated without a differentiable structure, using only distances and volumes, and the consequences which such estimates bring. Many important theorems from Riemannian geometry (volume growth estimates, spectral gaps, diameter bounds, factorization results, : : :) extend naturally to metric spaces with appropriate curvature bounds.These spaces also turn out to be the natural settings in which to analyze certain deterministic and probablistic dynamics modelling physical processes such as heat flow, chemical reactions, population spreading, and Markov chains. We touch on the consequences of bounds of sectional curvature type, before focusing on to the more subtle case where such bounds hold only in the Ricci (i.e. averaged) sense, a subject of intense and ongoing investigation.
We expose the necessary ideas from optimal transportation, which in particular allow more complicated spaces to be decomposed into collections of weighted one-dimensional metric spaces (`needles'). We discuss RCD(K;N) spaces and Gigli's geometric splitting theorem. Recalling that the Einstein _eld equation for gravity can be formulated in terms of Ricci curvature, if time and audience interests permit we shall explore how similar ideas can be used to formulate a nonsmooth theory of gravity still admitting key results such as the Hawking singularity theorem.
This course is offered in conjunction with the Fall 2026 Fields thematic semester on Optimal Transport in the Natural Sciences and Statistics.
Prerequisites:
Some familiarity with measure theory is assumed. Student familiarity with Differential geometry (Riemannian or Lorentzian) can provide useful intuition but is by no means essential.
References:
Ambrosio, Gigli and Savare: Gradient Flows in Metric Spaces and in the Space of Probability
Measures. Birkhäuser 2005.
Burago, Burago and Ivanov: A Course in Metric Geometry. AMS GSM #33, 2001.
Alexander, Kapovitch, and Petrunin: An Invitation to Alexandrov Geometry Spring 2019.
Cavalletti and Mondino, Optimal transport in Lorentzian synthetic spaces, synthetic timelike
Ricci curvature lower bounds and applications, Camb. J. Math. 12 (2024) 417534.
Gigli, The splitting theorem in non-smooth context arXiv:1302.5555
McCann, A synthetic null energy condition, Commun. Math. Phys. 405 (2024) 38:1-24.
Villani: Optimal Transport: Old and New, Springer-Verlag 2009.
MAT1502HS
Topics in Geometric Analysis: Variational methods in Analysis
This is a basic introduction to variational methods, as they appear in Functional Analysis, PDE, and Geometry. Topics include necessary conditions for minima, the Direct Method, symmetrization techniques, and applications (taking into account the interests of the audience).
MAT1525HS
Topics in Inverse Problems and Image Analysis: Variational Methods in Imaging and Generative Neural Networks
This course aims to provide the analytic tools for understanding some spectacularly successful generative neural networks from a mathematical point of view. Examples will include Wasserstein GANs and score-based diffusion models. We will spend much of the time on the requisite background from convex analysis, paradigmatic variational problems from image analysis, basics of optimal transport theory, stochastic differential equations and gradient flows in metric spaces.
Prerequisites:
asic real analysis and functional analysis will be helpful.
Evaluation:
20% attendance, 40% oral presentation of your project, 40% written report of the project.
Recommended References:
“Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling” by F. Santambrogio, Birkhauser, 2015.
“Gradient flows in Metric Spaces and in the Space of Probability Measures”, by L. Ambrosio, N. Gigli and G. Savare, Birkhauser, 2008.
“Wasserstein GANs with Gradient Penalty Compute Congested Transport” by T. Milne and A. Nachman, PMLR, pp. 103–129, 2022.
“Diffusion Models: A Comprehensive Survey of Methods and Applications”, by L. Yang, Z. Zhang, Y. Song, S. Hong, R. Xu, Y. Zhao, Y. Shao, W. Zhang, B. Cui, and M.-H. Yang, ACM Computing Surveys (56) pp 1–39, (2023).
“Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance”, by H. Wu, J. Kohler, and F. Noe”, Advances in Neural Information Processing Systems, (2022).
MAT1800HF
Methods of applied mathematics
The goal of this course is to introduce classical mathematical methods with applications to the analysis of nonlinear partial differential equations.
We will start with fundamental techniques for exponential integrals. In particular, the Laplace method, the method of stationary phase for oscillatory integrals and the steepest descent. We will apply these techniques to weakly diffusive shock waves and the Burger-Hopf equation, linear dispersive waves, as well as the Schrödinger equation and its semi-classical limit.
We will then present methods for asymptotics of solution of ODEs with respect to parameters, in particular the WKB (Wentzel-Kramers-Brillouin) approximation, the multiple-scale method and their application to weakly nonlinear waves, the Korteweg-de Vries equation and the nonlinear Schrödinger equation.
Finally we will introduce the inverse scattering method for integrable systems which include a large class of nonlinear equations. These equations have numerous features in common, one being the existence of solitons. Solitons are stable localized waves which interact elastically with each other.
The prerequisites are
Courses in ODEs, PDEs, Complex variables (preferably advanced undergraduate levels
Reference book :
Peter Miller, Applied Asymptotic Analysis, Amer. Math. Soc., 2006.
MAT1845HF
Dynamical Systems
M. Yampolsky
(View Timetable)
TBA